Question

graph each equation.

1. \\(\dfrac{x^2}{4} + \dfrac{y^2}{9} = 1\\)

Explanation:

Step1: Identify the conic section

The equation $\frac{x^2}{4}+\frac{y^2}{9} = 1$ is in the standard form of an ellipse, $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$ (where $a > b$ for a vertical major axis). Here, $a^2 = 9$ so $a = 3$, and $b^2=4$ so $b = 2$.

Step2: Find the vertices and co - vertices

• For the major axis (along the y - axis since $a$ is under $y^2$), the vertices are at $(0,\pm a)=(0,\pm3)$.
• For the minor axis (along the x - axis), the co - vertices are at $(\pm b,0)=(\pm2,0)$.

Step3: Plot the points

• Plot the vertices $(0, 3)$ and $(0, - 3)$ on the y - axis.
• Plot the co - vertices $(2,0)$ and $(-2,0)$ on the x - axis.

Step4: Draw the ellipse

Connect the plotted points smoothly to form the ellipse. The ellipse will be centered at the origin $(0,0)$ (since there are no shifts in the $x$ or $y$ terms in the standard form), with a vertical major axis of length $2a = 6$ and a horizontal minor axis of length $2b=4$.

To graph the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$:

1. Locate the center at $(0,0)$.
2. Mark the vertices at $(0, 3)$ and $(0, - 3)$ (on the y - axis) and the co - vertices at $(2,0)$ and $(-2,0)$ (on the x - axis).
3. Draw a smooth curve connecting these points to form the ellipse.

(Note: Since the question asks to graph the equation, the final answer is the graphical representation as described above. If we were to describe the key points for plotting, the main points are the center $(0,0)$, vertices $(0,\pm3)$ and co - vertices $(\pm2,0)$ which are used to sketch the ellipse.)

Answer:

The graph is an ellipse centered at the origin \((0,0)\) with vertices at \((0, 3)\), \((0, - 3)\) and co - vertices at \((2,0)\), \((-2,0)\). The ellipse is drawn by connecting these points smoothly, with a vertical major axis (length 6) and a horizontal minor axis (length 4).